In this section, we present some definitions and auxiliary results which will be needed in the sequel.

Denote by

the Banach space of continuous functions

, with the usual supremum norm

Let

be the Banach space of measurable functions

which are Bochner integrable, equipped with the norm

Let

be the Banachspace of measurable functions

which are bounded, equipped with the norm

Let
be the space of functions
, whose first derivative is absolutely continuous.

Moreover, for a given set

of functions

let us denote by

Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness.

Definition 2.1 (see [27]).

Let

be a Banach space and

the bounded subsets of

. The Kuratowski measure of noncompactness is the map

defined by

Properties

The Kuratowski measure of noncompactness satisfies some properties (for more details see [27]).

(a)
is compact (
is relatively compact).

(b)
.

(c)
.

(d)

(e)

(f)
.

Here
and
denote the closure and the convex hull of the bounded set
, respectively.

For completeness we recall the definition of Caputo derivative of fractional order.

Definition 2.2 (see [17]).

The fractional order integral of the function

of order

is defined by

where

is the gamma function. When

, we write

where

for
, and
as
.

Here
is the delta function.

Definition 2.3 (see [17]).

For a function

given on the interval

, the Caputo fractional-order derivative of

, of order

is defined by

Here
and
denotes the integer part of
.

Definition 2.4.

A map
is said to be Carathéodory if

(i)
is measurable for each

(ii)
is continuous for almost each

For our purpose we will only need the following fixed point theorem and the important Lemma.

Theorem 2.5 (see [31, 33]).

Let

be a bounded, closed and convex subset of a Banach space such that

, and let

be a continuous mapping of

into itself. If the implication

holds for every subset
of
, then
has a fixed point.

Lemma 2.6 (see [32]).

Let

be a bounded, closed, and convex subset of the Banach space

, G a continuous function on

and a function

satisfies the Carathéodory conditions, and there exists

such that for each

and each bounded set

one has

If

is an equicontinuous subset of

, then